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Structure of a Morse form foliation on a closed surface in terms of genus. (English) Zbl 1223.57022
Let \(\omega\) be a closed \(1\)-form with Morse singularities on a genus \(g\) closed surface \(M\). The form \(\omega\) defines a foliation \(\mathcal{F} _{\omega}\) on \(M\backslash Sing\;\omega\) and a singular foliation \(\overline{\mathcal{F}}_{\omega}\) on the whole \(M\). A leaf \(\gamma \in\mathcal{F}_{\omega}\) is compactifiable if \(\gamma\cup Sing\;\omega\) is compact. Non-compactifiable leaves of \(\mathcal{F}_{\omega}\) form minimal components \(\mathcal{C}_{j}^{\min};\) each such leaf is dense in its minimal component. If \(\gamma\) is a compact singular leaf, a small closed tubular neighborhood of \(\gamma\) is denoted by \(V(\gamma)\).
The main result of this work is the relation \(c(\omega)+\sum_{\gamma}g(V(\gamma))+g(\bigcup \overline{\mathcal{C}_{j}^{\min}})=g,\) where \(c(\omega)\) is the number of homologically independent compact leaves. This result allows the author to prove a criterion for compactness of \(\overline{\mathcal{F}}_{\omega}\), to estimate the number of its minimal components and to give an upper bound on the rank \(rk\;\omega,\) in terms of genus.

57R30 Foliations in differential topology; geometric theory
58K65 Topological invariants on manifolds
Full Text: DOI
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